the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
Various facets of intermolecular transfer of phase coherence by nuclear dipolar fields
Philippe Pelupessy
It has long been recognized that dipolar fields can mediate intermolecular transfer of phase coherence from abundant solvent to sparse solute spins. Generally, the dipolar field has been considered while acting during prolonged freeprecession delays. Recently, we have shown that transfer can also occur during suitable uninterrupted radio frequency pulse trains that are used for total correlation spectroscopy. Here, we will expand upon the latter work. First, analytical expressions for the evolution of the solvent magnetization under continuous irradiation and the influence of the dipolar field are derived. These expressions facilitate the simulations of the transfer process. Then, a pulse sequence for the retrieval of highresolution spectra in inhomogeneous magnetic fields is described, along with another sequence to detect a transfer from an intermolecular doublequantum coherence. Finally, various schemes are discussed where the magnetization is modulated by a combination of multiple selective radio frequency pulses and pulsed field gradients in different directions. In these schemes, the magnetization is manipulated in such a way that the dipolar field, which originates from a singlespin species, can be decomposed into two components. Each component originates from a part of the magnetization that is modulated in a different direction. Both can independently, but simultaneously, mediate an intermolecular transfer of phase coherence.
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In liquidstate NMR, the magnetization of an abundant or a highly polarized spin species affects the evolution of the density operator through radiation damping (RD) (Suryan, 1949) and through the dipolar field (DF) (Dickinson, 1951). RD stems from the radio frequency (rf) field caused by the current that the transverse magnetization induces in an rf coil (Bloembergen and Pound, 1954). The DF describes the direct contributions of the longitudinal and transverse nuclear magnetization components to the static field B_{0} and to a perpendicular rf field B_{1}, respectively, through dipolar interactions. It is also known as the DDF, which, in older works, stood for the dipolar demagnetizing field (Deville et al., 1979) but, in more recent works, has been redefined as the distant dipolar field (Ahn et al., 1998). Both RD and the DF can be incorporated into a modified set of Bloch equations (Bloom, 1957; Deville et al., 1979). Since RD results in a field with only a rapidly oscillating transverse component, its effect on other resonances is usually limited to nearby frequencies. When the chemical shift differences are removed from the effective Hamiltonian by suitable pulse sequences, the effects of RD extend over a much wider range of frequencies (Pelupessy, 2022a). Conversely, the DF has a longitudinal component which causes a shift in the precession frequencies of all nuclei that possess a spin (Edzes, 1990). Striking effects are observed when the magnetization of an abundant spin species depends on its spatial position, often as a result of a pulsed field gradient (PFG). These nontrivial effects include multiple spin echoes in twopulse experiments (Bernier and Delrieu, 1977; Bowtell et al., 1990) and intermolecular multiple quantum crosspeaks in COSYlike (correlation spectroscopy) sequences. These peaks can stem from like (He et al., 1993) or unlike (Warren et al., 1993) spins. The phase coherence of abundant spins can also be transferred by the DF during pulse trains that are commonly used in homonuclear total correlation spectroscopy (TOCSY) (Pelupessy, 2022b). As with RD in these type of experiments, the small transverse component of the DF plays an important role even if chemical shift differences are large.
In this work, several aspects of the transfer that is mediated by the DF and that occurs during continuous pulse trains will be explored: broadband inphase transfer and highresolution spectra can be obtained in inhomogeneous B_{0} fields in a fashion similar to the HOMOGENIZED (homogeneity enhancement by intermolecular zeroquantum detection; Vathyam et al., 1996) and related experiments (Lin et al., 2013). In addition, a transfer of phase coherence can be realized from intermolecular doublequantum (DQ) coherences involving the abundant solvent and the sparse solute spins. Finally, experiments where the magnetization is modulated in more complex ways by applying several PFGs in combination with selective rf pulses will be discussed.
2.1 The evolution of the magnetization during rf pulse trains
The following theory, originally developed by Deville et al. (1979), applies to a homonuclear spin system that contains an abundantspin species A and a sparsespin species S (both having a spin of $\mathrm{1}/\mathrm{2}$), where the magnetization of the spins A has been modulated in a manner that it averages out over the effective sample volume (Warren et al., 1995, expanded the theory for the case where this condition is not met). Moreover, the spatial variations must be in a single direction s. These modulations are usually induced by a field gradient which is oriented at an angle θ_{G} with respect to B_{0}, by convention along the z axis, so that $\mathrm{cos}{\mathit{\theta}}_{\mathrm{G}}=\widehat{s}\cdot \widehat{z}$. The DF can then be characterized by an angular frequency ω_{d}, defined as follows:
with γ being the gyromagnetic ratio of the A and S spins, μ_{0} being the vacuum permeability, and ${M}_{\mathrm{eq}}^{A}$ being the magnitude of the magnetization of the A spins at equilibrium. In the rotating frame, the evolution of the magnetization vectors of both A and S spins is governed by the modified Bloch equations (Deville et al., 1979; Bowtell et al., 1990; Enss et al., 1999):
with i being either A or S, ${\mathit{\omega}}_{\mathrm{1}x/y}$ being the (timedependent) rf field, and ω_{0} being the offset from the carrier frequency. The magnetization components, written in lowercase m, are normalized with respect to the equilibrium amplitudes. These equations are local (as a result of the modulations being only in one direction; Deville et al., 1979) so that the evolution can be calculated separately for each position in the sample. Typically, the rf term does not appear in these equations since, previously, the DF has been considered predominantly during freeprecession delays. The explicit position and time dependence of the variables have been omitted in these equations. A set time t=T will be indicated in brackets. Neither relaxation nor molecular motions by diffusion or convection have been taken into account.
The set of coupled equations in Eq. (2) is nonlinear when i=A, while for the sparse spins i=S, the magnetization of the A spins ${m}^{A}=({m}_{x}^{A},{m}_{y}^{A},{m}_{z}^{A})$ is the source of a timedependent field $\mathit{\gamma}{B}_{\mathrm{d}}=({\mathit{\omega}}_{\mathrm{d}}{m}_{x}^{A}/\mathrm{2},{\mathit{\omega}}_{\mathrm{d}}{m}_{y}^{A}/\mathrm{2},{\mathit{\omega}}_{\mathrm{d}}{m}_{z}^{A})$. Hence, the evolution of the magnetization of the S spins ${m}^{S}=({m}_{x}^{S},{m}_{y}^{S},{m}_{z}^{S})$ can be calculated straightforwardly from the trajectory of m^{A}. In Pelupessy (2022b), the evolution of m^{A} during the rf pulse trains has been obtained by numerical integration of the nonlinear set of Eq. (2). In this work, a DIPSI2 (decoupling in presence of scalar interactions; Rucker and Shaka, 1989) pulse train is always applied at the resonance frequency of the A spins, in which case the trajectory of m^{A} can be approximated analytically as follows: in Appendix A, it is derived that, for a strong constant onresonance rf field ($\left{\mathit{\omega}}_{\mathrm{1}}\right\gg \left{\mathit{\omega}}_{\mathrm{d}}\right$) along the x axis, m^{A} rotates around x with an angular frequency of ${\mathit{\omega}}_{\mathrm{1}x}\mathrm{3}{m}_{x}^{A}\left(\mathrm{0}\right){\mathit{\omega}}_{\mathrm{d}}/\mathrm{4}$:
Many sequences used for decoupling or magnetization transfer consist of a repetitive cycle of phasealternated pulses along one axis (in this work, this is assumed to be between +x and −x), as is the case for, for example, DIPSI2, Waltz16 (Shaka et al., 1983), and GARP (globally optimized alternatingphase rectangular pulses, Shaka et al., 1985). Typically, the pulses are constant in amplitude but may differ in duration. By design, the integral of the rf field ∫ω_{1x}dt averages to zero over one cycle. Consequently, the contribution of ω_{1x} vanishes after each full cycle if Eq. (3) governs the evolution during all pulses in the cycle. In Fig. 1a–c, the validity of this approximation is tested for DIPSI2 by comparing the trajectories predicted by Eq. (3) with exact numerical simulations as described in Pelupessy (2022b). When the initial magnetization is oriented far away from the x axis, the trajectories diverge at longer irradiation times t_{p} of the DIPSI2 pulse train. These differences stem from small contributions of the oscillating terms in Eq. (A5) due to the rapid switching of rf phases, which accumulate as the number of cycles increases. For shorter times t_{p}<0.4 s, the curves obtained by both methods agree quite well.
For a continuous unmodulated rf field, the trajectories of m^{A} calculated with the two methods are indistinguishable, while for a GARP pulse train, where the phases are alternated 2.5 times more frequently than for DIPSI2 with the same rf amplitude, the deviations are much larger (see the Supplement). Although one might be tempted to attribute this result to the higher phaseswitching rate, the details of the sequence are also important. For example, the use of WALTZ16, with a phaseswitching rate that is 1.6 times more frequent than for DIPSI2, results in a perfect match between the trajectories calculated with the two methods. The elementary block of WALTZ16 consists of three pulses, $\mathrm{90}{{}^{\circ}}_{x}\mathrm{180}{{}^{\circ}}_{x}\mathrm{270}{{}^{\circ}}_{x}$, i.e., a 180^{∘} pulse flanked by two pulses whose sum is 360^{∘}. These angles make the oscillating terms in Eq. (A5) run over exactly 1 and 2 full rotations.
Figure 1d shows simulations of the intermolecular transfer of phase coherence due to the DF from spins A to spins S for a gradientselected selective TOCSY experiment (Dalvit and Bovermann, 1995; Pelupessy, 2022b). As in Fig. 1a–c, the solid lines have been obtained with the trajectory of m^{A}, calculated by numerical integration of the nonlinear coupled differential Eq. (2), as described in Pelupessy (2022b), while the dashed lines have been obtained with the trajectory predicted by Eq. (3). For the latter simulations (see the supporting information for the code), Eq. (3) with ω_{1x}=0 has been used to calculate the evolution of m^{A} between the DIPSI2 cycles, while, within each cycle, the trajectory was assumed to be solely determined by the rf field. The dashed lines that can be calculated very rapidly are barely distinguishable from the solid ones that require the more laborious simulations of Pelupessy (2022b).
Effects of RD may mask or dampen those of the DF since the timescale in which RD occurs can be more than 1 order of magnitude shorter (Desvaux, 2013). However, in the experiments shown in this work, the former are suppressed by dephasing the A spins with a 90^{∘} pulse followed by a PFG, which allows us to focus on effects of the DF.
2.2 A qualitative description of the experiments
In the next section, several TOCSYlike experiments, where the DF mediates an intermolecular transfer of phase coherence, will be discussed. While the above theory will be used to simulate the transfer, the formalism developed by Warren and coworkers (Lee et al., 1996) will be employed to guide the experimental design. For this, the hightemperature approximation needs to be abandoned by taking into account higherorder terms in the expansion of the density matrix:
Since intermolecular interactions between sparse spins S can be neglected, the index i runs over all abundant spins A. In the experiments explored in this work, the equilibrium density operator evolves during a preparation period – before the DIPSI2 irradiation – under the influence of rf pulses and PFGs. In this period, the DF will be neglected either because of its brevity or because the relevant part of the density operator commutes with the effective dipolar Hamiltonian. The intermolecular dipolar interactions between the A and S spins need to be tracked. The effective intermolecular dipolar Hamiltonian during the DIPSI2 sequence is given by Kramer et al. (2001):
The transfer induced by the DF can be evaluated by calculating the commutator of the density matrix and this Hamiltonian. The precise coefficients have been omitted in these equations since they are not needed for a mere qualitative description. Likewise, only the minimal number of spins A, which may cause an observable signal on the S spins, will be considered. In this work, it is sufficient to account for twospin operators containing one A and one S spin since the different transfers involve only operators containing singlequantum (SQ) terms of the A spins (the experiment described in Sect. 3.2 involves a DQ coherence which is a product of SQ terms of the A and S spins). For a qualitative description of the experiments, SQ operators containing only a single A spin are sufficient. For quantitative calculations, SQ operators containing multiple A spins (such as ${A}_{+}^{\mathrm{1}}{A}_{}^{\mathrm{2}}{A}_{+}^{\mathrm{3}}$) are also needed (Lee et al., 1996). The A spins are assumed to have the same spatial coordinates as the S spins. These simplifications preclude a precise description of the evolution due to the DF; neither the correct amplitude nor the angular dependence can be predicted. Nevertheless, it allowed good insight into the original experiment where a DIPSI2 sequence was used to transfer the coherence between A and S spins and even provided a close estimate of the ratio of the initial rates of transfer into the different coherence orders (Pelupessy, 2022b).
In Pelupessy (2022b), we demonstrated that the DF can efficiently mediate an intermolecular transfer of phase coherence during DIPSI2 pulse trains. In anterior work, where the DF acted during freeprecession delays, the change of coherence order needed to be achieved by rf pulses. On the contrary, the effective dipolar Hamiltonian during the DIPSI2 irradiation allows for a change in coherence order by the DF. The transfer from a +1 to a −1 coherence was shown to be particularly efficient. This coherence order pathway refocuses B_{0} inhomogeneities. In Sect. 3.1, the original experiment is adapted to obtain highresolution spectra in inhomogeneous fields in a HOMOGENIZEDlike (Vathyam et al., 1996) fashion. In Sect. 3.2, an alternative coherence selection pathway will be investigated: the transfer mediated by the DF from an intermolecular DQ coherence to z magnetization. When multiple PFGs in different directions combined with several rf pulses are applied, the modulation pattern of the magnetization can become more complex. In Sect. 3.3, the influence of this kind of modulation on the intermolecular transfer will be investigated.
3.1 Highresolution spectra in inhomogeneous fields
Figure 2a depicts an adaptation of the selective TOCSY pulse sequence (Pelupessy, 2022b) with DIPSI2 rf irradiation used to transfer the phase coherence from the abundant solvent spins A to the solute spins S, which includes (1) a WATERGATE spin echo (Piotto et al., 1992) to refocus the chemical shift evolution after the DIPSI2 irradiation and to achieve solvent suppression and (2) an indirect evolution period before the TOCSY pulse train to record highresolution spectra in inhomogeneous B_{0} fields, similarly to experiments by Huang et al. (2010), which used the DF combined with spin echo correlation spectroscopy (SECSY, Nagayama et al., 1979). In Fig. 2b, a 1D spectrum is obtained with t_{1}=0 in a homogeneous B_{0} field of 2 mM sucrose and 0.5 mM sodium 3(trimethylsilyl)propane1sulfonate (DSS) in a 90 % $/$ 10 % mixture of H_{2}O $/$ D_{2}O (the same sample is used throughout this work). This sample is a standard to test water suppression, with a welldispersed distribution of resonances between 0 and 6 ppm. The inset, which shows part of the spectrum, highlights the small phase distortions at long DIPSI2 irradiation times (> 100 ms), which cannot be simultaneously corrected for all resonances by a linear phase correction. The ^{13}C satellites on both sides of the methyl protons (marked with asterisks) correspond to a concentration of 22.5 µM.
The results of a 2D experiment in an inhomogeneous B_{0} field are shown in Fig. 3a (the line width at half height measured on the water resonance was about 225 Hz). The coherence selection pathway below the sequence in Fig. 2 shows that B_{0} inhomogeneities that have evolved in t_{1} should be refocused during the direct dimension at the time t_{2}=t_{1}, which leads to the skewed line shapes. No corrections or special processing protocols were applied to remove phase twists in the spectrum. The absence of those are due to constructive interference of neighboring peaks (Sect. 6.5.2 of Ernst et al., 1992). Moreover, even if present, phase twists hardly perturb the results (see supporting information). In the indirect t_{1} dimension, the spectrum needs to cover the inhomogeneously broadened line shape (here, 1 ppm was used). On the right (Fig. 3b), a sliding window function was applied in the direct t_{2} dimension, and subsequently, the spectrum was sheared so that the elongated ridges appear perpendicular to the ν_{2} axis. The window function W_{i} consisted of zeroes for all time points, except for a narrow range of points $k=\mathit{\{}d+o,d+o\mathit{\}}$ where the intensities were multiplied by
where bw_{2} is the bandwidth in the direct dimension (the inverse of the time increment), and the offset o is the integer nearest to bw_{2}t_{1}. The higher the integer value n, the closer it is to a rectangular profile. The broader the inhomogeneous line, the sharper the echo and, consequently, the smaller the range 2d. This function is identical to the amplitude modulation of wideband, uniformrate, and smoothtruncation pulses (WURST; Kupce and Freeman, 1995). The variables d=300 and n=1 were optimized empirically (the value of n only slightly affects the result). Application of this window function resembles chunk selection in pure shift NMR (Zangger and Sterk, 1997).
The 1D spectrum in Fig. 3c corresponds to the sum of the middle 112 rows (of a total of 512) of the spectrum of Fig. 3b. On the side panels, parts of this spectrum (Fig. 3d1 and e1, corresponding to a very crowded region and to a weak and complex multiplet) are compared with the results of a pulseacquire experiment preceded by saturation of the solvent signal in a homogeneous B_{0} field (Fig. 3d2 and e2). Fig. 3d3 and e3 show parts of the Fourier transform of the first increment (t_{1}=0). The enlargements of Fig. 3d1 and e1 closely resemble those of Fig. 3d2 and e2. In the lower spectra, slight distortions due to scalarcoupling evolution during the WATERGATE spin echo are visible, and the relative peak intensities do not exactly match those in the center spectra.
The experiment presented in this section is capable of delivering broadband, inphase, wellresolved spectra with only small distortions. It shares these characteristics with the experiment of Fugariu et al. (2017), which adds an adiabatic spinlock to the method of Huang et al. (2010) in order to avoid scalarcoupling evolution during the transfer by the DF but keeps the same effective dipolar Hamiltonian as the one during free precession and achieves the changes in coherence order solely by rf pulses. Since the effective Hamiltonian is scaled during the DIPSI2 pulse train, the transfer is slower compared to sequences which rely on prolonged freeprecession delays. On the other hand, the entire (phasemodulated) magnetization of the solvent contributes to the transfer. If the solute relaxation rates are higher than the ones of the solvent (as is often the case), the new sequence should be advantaged since the losses depend on a mix of longitudinal and transverse relaxation rates. Moreover, contributions from conformational exchange to the relaxation will be attenuated. In vivo, its use may be limited due to the prolonged duration of rf irradiation.
3.2 Doublequantum transfer
In the original CRAZED sequence (COSY revamped with asymmetric zgradient echo detection) of Warren et al. (1993), intermolecular DQ coherences are converted to SQ ones by a 90^{∘} pulse. The DF then converts these multiplespin SQ coherences to observable onespin SQ coherences. The appropriate coherence order pathway needs to be selected by a judicious choice of PFGs, in this case a 1:2 ratio of the areas of the PFGs before and after the 90^{∘} mixing pulse. This rf pulse is essential because the dipolar Hamiltonian does not allow for a change in coherence order. As seen in the previous section, this constraint does not apply to the effective Hamiltonian of Eq. (5). Hence, it may be interesting to explore the effect of the DF on intermolecular DQ coherences during DIPSI2 irradiation.
After the first 90^{∘} pulse of the sequence in Fig. 4a, the magnetization of the A and the S spins is dephased by a PFG G_{a}. The commutator of ${H}_{AS}^{\mathrm{eff}}$ of Eq. (5) with the lowestorder term in the expansion of the density operator, which contains a product of S and A spin operators, results in the following:
where c_{a} and s_{a} stand for cos (τ_{a}γG_{a}⋅r) and sin (τ_{a}γG_{a}⋅r), with τ_{a} being the effective duration of G_{a} and r being the spatial position. In general, for arbitrary numbers p and q, c_{pa+qb} stands for $\mathrm{cos}(p{\mathit{\tau}}_{a}\mathit{\gamma}{G}_{a}\cdot r+q{\mathit{\tau}}_{b}\mathit{\gamma}{G}_{b}\cdot r)$. Thus, the DIPSI2 irradiation leads to a transfer of a DQ coherence involving the A and S spins to longitudinal magnetization. A purge gradient followed by a 90^{∘} rf pulse and a final PFG +2G_{a} or −2G_{a} (pG_{a} stands for a PFG along the direction of G_{a} with an area p times as large) allows one to detect the longitudinal term of the sparse S spins. A phase difference ϕ due to chemical shift evolution causes a phase shift ϕ in the signal and additional longitudinal components which are not modulated by the gradient.
In Fig. 5a, the intensity, ${m}_{xy}=\sqrt{{m}_{x}^{\mathrm{2}}+{m}_{y}^{\mathrm{2}}}$, of the methyl proton resonance of DSS is plotted as a function of the orientation angle ${\mathrm{\Theta}}_{{G}_{a}}$ of the PFG G_{a} for a DIPSI2 irradiation time of about 100 ms. The line that goes through the points corresponds to a function ${m}_{xy}^{\mathrm{0}}(\mathrm{3}{\mathrm{cos}}^{\mathrm{2}}{\mathrm{\Theta}}_{{G}_{a}}\mathrm{1})/\mathrm{2}$, where ${m}_{xy}^{\mathrm{0}}$ is the intensity recorded with the PFG oriented parallel to B_{0}. The line crosses zero at the socalled magic angle Θ=54.74^{∘}. Although, by definition, all intensities are positive, for clarity, the intensities of signals that point in opposite directions when phased identically are plotted with opposite signs. The theoretical curve should match the experimental points only at short irradiation times (i.e., in the linear regime of the buildup).
In Fig. 5b, the buildup of intensities is plotted as a function of the irradiation time t_{p} for orientations that are parallel (${\mathrm{\Theta}}_{{G}_{a}}=\mathrm{0}{}^{\circ}$, blue squares) and perpendicular (${\mathrm{\Theta}}_{{G}_{a}}=\mathrm{90}{}^{\circ}$, blue circles) with respect to B_{0}. The dotdashed blue lines are simulations of these experiments using the approximate solution of the nonlinear Bloch Eq. (2) for the evolution of m^{A}. The intensities for ${\mathrm{\Theta}}_{{G}_{a}}=\mathrm{90}{}^{\circ}$ at longer irradiation times exceed half of those for ${\mathrm{\Theta}}_{{G}_{a}}=\mathrm{0}{}^{\circ}$. This is because, when the magnitude of ω_{d} is divided by a factor of 2, the transfer is not half as efficient but rather twice as slow (of course, a slower transfer renders the experiment more sensitive to losses due to relaxation and diffusion). The red crosses are the intensities obtained with the experiment described in Pelupessy (2022b) (i.e., the experiment of Fig. 2a but without refocusing pulses, solvent suppression, and an indirect evolution period) for ${\mathrm{\Theta}}_{G}=\mathrm{0}{}^{\circ}$ and a coherence pathway selection $+\mathrm{1}\to \mathrm{1}$. For clarity, the latter intensities have been divided by 2 since the (simplified) commutator formalism predicts that the initial slopes should differ by a factor 2. This factor applies only for the initial slopes. Neglecting relaxation and diffusion, the transfer reaches a theoretical maximum of ${m}_{xy}^{S}=\mathrm{0.88}$ at t_{p}≈450 ms for the experiment of Fig. 2 and of ${m}_{xy}^{S}=\mathrm{0.32}$ at t_{p}≈320 ms for the experiments of Fig. 4.
In the sequence of Fig. 4b, the solvent and solute spins are labeled by different gradients, and the chemical shifts are refocused. The A spins are dephased by 4G_{a} and the S spins by 4G_{b}. During the DIPSI2 irradiation, the DF induces the following transfer:
As in the previous sequence, only singlespin SQ terms of the A spins suffice for a qualitative description. The first term on the right can be recovered after a 90^{∘} pulse by the sum of the two PFGs, $\pm (\mathrm{4}{G}_{a}+\mathrm{4}{G}_{b})$. The solvent senses only G_{a} before the DIPSI2 irradiation so that no additional measures need to be taken to suppress its signal after the DIPSI2 irradiation, and, consequently, the echo before signal detection is shorter than the one in Fig. 2. The characteristic frequency ω_{d} of the DF depends only on the orientation of G_{a}.
In Fig. 5c, spectra obtained with G_{a} along the z axis and with G_{b} along the x axis for several DIPSI2 irradiation times are displayed. The buildup curves of three resonances (the methyl resonance of DSS and two sucrose resonances) are plotted in Fig. 5d. The buildup curves are very similar at short times and start to diverge at longer times, probably due to differences in relaxation rates.
3.3 Mixed modulations
When applying multiple rf pulses interleaved with PFGs in different directions, the modulation of the magnetization can become rapidly very convoluted. Often, PFGs serve as purging gradients, and the parts of the density operator that are dephased by these PFGs can be discarded. However, magnetization which is modulated in complex patterns can also be implicated in a transfer of phase coherence by the DF, although the conditions for the validity of the theory described in Sect. 2.1 may not strictly apply anymore. In this section, this type of more intricate modulations will be investigated. Several schemes will be presented where the density operator is prepared in different ways and where the transfer from a +1 to a −1 coherence order is recorded.
At the start of the sequence of Fig. 6a, a PFG G_{a} sandwiched between two selective 90^{∘} pulses modulates the amplitude of the magnetization of the abundant A spins. It is followed by a purging PFG and another selective pulse and PFG G_{b} before DIPSI2 irradiation. Neglecting the parts of the density operator which are dephased by G_{c}, the transfer due to the DF that occurs during the DIPSI2 irradiation can be described as follows:
Using the relations ${c}_{a}{c}_{b}=({c}_{b+a}+{c}_{ba})/\mathrm{2}$ and ${c}_{a}{s}_{b}=({s}_{b+a}+{s}_{ba})/\mathrm{2}$, the DF can be decomposed into two fields, one originating from the half of the magnetization of the A spins that is dephased by G_{b}−G_{a} and the other from the half that is dephased by G_{b}+G_{a}. If the gradients G_{b} and G_{a} are oriented either parallel or perpendicular to B_{0}, both fields are characterized by the same value of ω_{d}. With the first two pulses applied selectively to the S instead of to the A spins, shown in Fig. 6b, the operator analysis remains the same so that Eq. (9) also describes the transfer in this case.
The experiments of Fig. 6a and b were performed with G_{b} along z and with G_{a} along x and a DIPSI2 irradiation time of 100 ms. The areas of the PFGs have been varied to change the angle ${\mathrm{\Theta}}_{{G}_{a}+{G}_{b}}$ of the vector addition of G_{a} and G_{b} with respect to B_{0} while keeping the amplitude constant. The transfer efficiency is plotted as a function of ${\mathrm{\Theta}}_{{G}_{a}+{G}_{b}}$ in Fig. 7a and b for the experiments of Fig. 6a and b, respectively. Only in the first experiment where all selective pulses are applied to H_{2}O is the typical (3cos ^{2}Θ−1) dependence observed. The direction of the spatial modulation of the A spins is important; this does not change with ${\mathrm{\Theta}}_{{G}_{a}+{G}_{b}}$ in the experiment of Fig. 6b since the first two pulses are applied on the S spins. While, for intramolecular twospin operators, it does not matter how the amplitude or phase modulation has been created, for intermolecular twospin operators, Eq. (9) does not provide the full picture. For a spin S at a given position, one has to consider the dipolar interactions with all spins A (Lee et al., 1996), whose spatial modulation is different for the two experiments. The $(\mathrm{3}{\mathrm{cos}}^{\mathrm{2}}{\mathrm{\Theta}}_{{G}_{a}+{G}_{b}}\mathrm{1})$ dependence, which is absent in the experiment of Fig. 6b, is restored in the sequence of Fig. 6c, where the second gradient G_{a} is put before the DIPSI2 block instead of behind, as shown in Fig. 7c.
For the experiment of Fig. 6a, Fig. 7d shows the transfer as a function of the DIPSI2 irradiation time at angles ${\mathrm{\Theta}}_{{G}_{a}+{G}_{b}}=\mathrm{15}{}^{\circ}$ (blue circles) and ${\mathrm{\Theta}}_{{G}_{a}+{G}_{b}}=\mathrm{80}{}^{\circ}$ (green triangles). The dotdashed curves are simulations that assume ω_{d} is constant throughout the irradiation time, which may not be entirely correct since the spatial pattern of the modulations of the magnetization of A will change over time (although the modulations will remain in the plane spanned by the vectors in the directions of G_{b}−G_{a} and G_{b}+G_{a}). The red crosses correspond to the experiment of Fig. 6c, with ${\mathrm{\Theta}}_{{G}_{a}+{G}_{b}}=\mathrm{15}{}^{\circ}$. The buildup is almost indistinguishable from the one in blue circles. The simulated curves for the latter experiment (here, ω_{d} is constant since the magnetization of A is modulated in one direction only) are almost identical to the previous ones, indicating that the assumption of a constant ω_{d}, as used for the simulations of experiments of Fig. 6a, is a reasonable approximation for these DIPSI2 irradiation times.
In the experiments above, the modulation of the magnetization was decomposed into two directions, not necessarily orthogonal, each at the same angle with respect to B_{0}. This is different in the following experiment. Just before the DIPSI2 irradiation in the pulse sequence of Fig. 6d, the twospin S_{z}A_{z} term of ρ_{eq} in Eq. (4) has evolved into the following:
A term containing S_{z}A_{z} (created by the second selective pulse) has been left out since it commutes with the effective dipolar Hamiltonian of Eq. (5) and will not lead to an observable signal. Equation (10) corresponds to a situation where half of the magnetization is modulated in the direction of G_{a}, and the other half is modulated in the direction of G_{b}. Experiments were performed with G_{a} being parallel and G_{b} being perpendicular to B_{0}, with different irradiation times. The blue circles in Fig. 8 correspond to the signal intensities of the methyl protons of DSS while using a PFG 2G_{a} before detection. The green squares correspond to intensities recorded using a PFG −2G_{b}. The signal intensities represented by the red crosses were recorded under the same circumstances as the second set of experiments (i.e., with −2G_{b} before acquisition), except that G_{a} was perpendicular to both G_{b} and B_{0}. The intensities of the latter experiments at longer times are slightly higher than the intensities of the green squares. This is likely caused by the larger perturbation of the longitudinal magnetization of the S nuclei that occurs when one of the modulations is parallel to the B_{0} field.
When the magnetization is modulated in only one dimension, the strength of the DF scales with the local value of m^{A}. This is not true anymore with the last scheme of this section since the DF can be decomposed into two different parts, each with its own characteristic value of ω_{d}. Remarkably, for a given position in the sample, the magnetization of the solute m^{S} may be affected by the DF; albeit, the solvent magnetization vanishes (m^{A}=0) at that position. Another counterintuitive feature is that, even if the vector addition of the PFGs after the first rf pulse and the vector addition of the PFGs after the second rf pulse both point along the magic angle with respect to B_{0}, it is still possible to observe a transfer of phase coherence. Hence, application of PFGs oriented along the magic angle does not always suffice to suppress the effects of the DF.
All experiments were acquired in a B_{0} field of 18.8 T (800 MHz proton frequency) at 290 K, with a probe equipped with coils to generate PFGs along three orthogonal axes. The rf amplitude ${\mathit{\omega}}_{\mathrm{1}}/\mathrm{2}\mathit{\pi}$ during the DIPSI2 pulse train was 8.33 kHz. The selective pulses on either the solvent or the methyl protons of DSS had Gaussian shapes and a length of 5 ms (for both the 90 and 180^{∘} pulses), except the two 90^{∘} pulses in the WATERGATE scheme of Fig. 2, which had a sinc profile and a duration of 2 ms. The shaped pulses that were applied on the equilibrium magnetization of the abundant spins A were calibrated separately to compensate for the effects of RD. All PFGs had smoothed square profiles and durations of 1 ms. G_{1}, G_{2}, and G_{3} indicate orthogonal gradient channels. The amplitudes of the PFGs were G_{a}=1.6, G_{b}=6.2, ${G}_{a+b}=\mathrm{7.8}$, and G_{c}=32 G cm^{−1} for the experiments of Figs. 2 and 3; G_{a}=7.8 and G_{c}=27 G cm^{−1} for Fig. 5a and b; G_{a}=1.95, G_{b}=1.95, G_{c}=27, and G_{d}=10 G cm^{−1} for Fig. 5c and d; $\Vert {G}_{a}+{G}_{b}\Vert =\mathrm{7.8}$ and G_{c}=32 G cm^{−1} for Fig. 7; and G_{a}=3.9 and G_{b}=3.9 G cm^{−1} for Fig. 8. For all experiments, 8192 complex points were acquired with a bandwidth of 12 ppm. All signal intensities that have been quantified have been normalized either to the spectrum of the methyl protons after a selective excitation or to a broadband pulseacquire experiment preceded by saturation of the solvent signal.
The 2D spectra of Fig. 3 have been acquired with 256 t_{1} increments, a bandwidth of 1 ppm, four scans per increment in the indirect dimension, and a repetition time of 11 s (in the supporting information, a similar experiment recorded with only one scan, a repetition time of 3 s, and a DIPSI2 irradiation time of 100 ms is shown). For these spectra, the experimental data have been doubled in each dimension by zero padding. The spectrum of Fig. 3a has been obtained by a 2D Fourier transform without any apodization. For the spectrum of Fig. 3b, the same data have been multiplied by the window function of Eq. (6) and subsequently Fourier transformed along the indirect t_{1} dimension. A firstorder phase correction along the direct t_{2} dimension, proportional to the ν_{1} position, was then applied to shear the spectrum followed by a Fourier transform along the same dimension.
In this work, we have investigated several aspects of the transfer of phase coherence by the dipolar field during rf irradiation sequences that have been developed for total correlation spectroscopy, in particular the DIPSI2 pulse train. Theoretical expressions for the evolution of the solvent spins under continuous rf irradiation have been derived, which permits efficient simulations of the transfer process. A remarkable feature is that, under these conditions, the DF can cause not only a transfer but also a change in coherence order. Nevertheless, the formalism developed by Warren and coworkers (Lee et al., 1996) can still be used – taking into account the effective Hamiltonian of Eq. (5) – to describe and design pulse sequences. An experiment for the acquisition of broadband, inphase, highresolution spectra in inhomogeneous fields has been presented. In this experiment, the transfer takes place from a SQ coherence of the abundant solvent spins to another SQ coherence of the sparse solute spins. Additionally, alternative coherence order pathways have been investigated: a DQ coherence involving both solvent and solute spins can be converted by the DF into longitudinal magnetization. Two pulse sequences to record this transfer have been introduced. In the first sequence, the DQ coherence is dephased by a PFG, while the SQ coherence, which is detected, is rephased with a PFG that is twice as strong. In the second sequence, the solvent and solute parts of the DQ coherence are dephased by two different PFGs. The SQ coherence is now rephased by the sum of these two PFGs. The latter sequence has the advantage of resulting in broadband, inphase spectra without the need for additional water suppression before acquisition. In the last part, more complex modulation patterns of the magnetization have been investigated. The importance of how the magnetization of the different spins has been modulated in intermolecular multiplespin operators has been explored with three closely resembling pulse sequences. Finally, by combining several pulsed field gradients and rf pulses, the DF has been tailored in such a manner that it can be decomposed into different components that have their own spatial modulation and, hence, can simultaneously bring about different transfer processes.
In the different sequences, DIPSI2 pulse trains have been applied. Other TOCSY mixing sequences can be used, although one has to take into account the different effective dipolar Hamiltonians that characterize these sequences (Kramer and Glaser, 2002). These effects could be suppressed by the use of tailored mixing sequences (Klages et al., 2007) or the use of PFGs along the magic angle. However, the results in Sect. 3.3 caution against the latter option for sequences with many pulses and PFGs. The effects of the transfer by the scalar couplings during the TOCSY sequence have not been taken into account. This is reasonable because either an uncoupled nucleus has been probed or because the solute coherences all had the same phase or were all along the z axis before the DIPSI2 irradiation, which minimizes the effects of the transfer by scalar couplings (Braunschweiler and Ernst, 1983). In sequences where the chemical shifts of the solute spins evolve before the TOCSY pulse train, a more complex behavior is expected.
Consider the case of a constant onresonance rf field. Without loss of generality, the rf field can be oriented along the x axis. Hence, for the abundant A spins, the set of Eq. (2) can be written as follows:
where $\mathit{\alpha}=\mathrm{3}{\mathit{\omega}}_{\mathrm{d}}/\mathrm{4}$. In terms of
Eq. (A1) becomes
Usually, m_{+} and m_{−} are defined in terms of transverse operators. This traditional definition could have been kept by transforming to a tilted frame. Switching to a rotating frame around the x axis,
one obtains
When the oscillating components can be neglected (i.e., when $\left{\mathit{\omega}}_{\mathrm{1}x}\right\gg \left\mathit{\alpha}\right$), the solution is a nutation around the x axis with an angular frequency of $\mathit{\alpha}{m}_{x}^{A}\left(\mathrm{0}\right)$, which, in the original frame, results in
The Python pulse program used for different simulations can be found in the Supplement.
The supplement related to this article is available online at: https://doi.org/10.5194/mr42712023supplement.
The author has declared that there are no competing interests.
Publisher’s note: Copernicus Publications remains neutral with regard to jurisdictional claims made in the text, published maps, institutional affiliations, or any other geographical representation in this paper. While Copernicus Publications makes every effort to include appropriate place names, the final responsibility lies with the authors.
I thank Geoffrey Bodenhausen for the careful reading and for correcting the paper and Kirill Sheberstov for fruitful discussions.
This paper was edited by Zhehong Gan and reviewed by Warren Warren, Malcolm Levitt, and Norbert Mueller.
Ahn, S., Lee, S., and Warren, W. S.: The competition between intramolecular J couplings, radiation damping, and intermolecular dipolar couplings in twodimensional solution nuclear magnetic resonance, Mol. Phys., 95, 769–785, https://doi.org/10.1080/002689798166404, 1998. a
Bernier, M. and Delrieu, J.: Measurement of Susceptibility of Solid He3 Along Melting Curve from 20 Mk down to Nuclear Ordering Temperature, Phys. Lett. A, 60, 156–158, https://doi.org/10.1016/03759601(77)904145, 1977. a
Bloembergen, N. and Pound, R.: Radiation Damping in Magnetic Resonance Experiments, Phys. Rev., 95, 8–12, https://doi.org/10.1103/PhysRev.95.8, 1954. a
Bloom, S.: Effects of Radiation Damping on Spin Dynamics, J. Appl. Phys., 28, 800–805, https://doi.org/10.1063/1.1722859, 1957. a
Bowtell, R., Bowley, R., and Glover, P.: Multiple Spin Echoes in Liquids in a High MagneticField, J. Magn. Reson., 88, 643–651, https://doi.org/10.1016/00222364(90)90297M, 1990. a, b
Braunschweiler, L. and Ernst, R.: Coherence Transfer by Isotropic Mixing – Application to Proton Correlation Spectroscopy, J. Magn. Reson., 53, 521–528, https://doi.org/10.1016/00222364(83)902263, 1983. a
Dalvit, C. and Bovermann, G.: PulsedField Gradient OneDimensional Nmr Selective Roe and Tocsy Experiments, Magn. Reson. Chem., 33, 156–159, https://doi.org/10.1002/mrc.1260330214, 1995. a
Desvaux, H.: Nonlinear liquidstate NMR, Prog. Nucl. Magn. Reson. Spectrosc., 70, 50–71, https://doi.org/10.1016/j.pnmrs.2012.11.001, 2013. a
Deville, G., Bernier, M., and Delrieux, J.: Nmr Multiple Echoes Observed in Solid He3, Phys. Rev. B, 19, 5666–5688, https://doi.org/10.1103/PhysRevB.19.5666, 1979. a, b, c, d, e
Dickinson, W.: The Time Average Magnetic Field at the Nucleus in Nuclear Magnetic Resonance Experiments, Phys. Rev., 81, 717–731, https://doi.org/10.1103/PhysRev.81.717, 1951. a
Edzes, H.: The Nuclear Magnetization as the Origin of Transient Changes in the MagneticField in Pulsed Nmr Experiments, J. Magn. Reson., 86, 293–303, https://doi.org/10.1016/00222364(90)902617, 1990. a
Enss, T., Ahn, S., and Warren, W. S.: Visualizing the dipolar field in solution NMR and MR imaging: threedimensional structure simulations, Chem. Phys. Lett., 305, 101–108, https://doi.org/10.1016/S00092614(99)003668, 1999. a
Ernst, R. R., Bodenhausen, G., and Wokaun, A.: Principles of nuclear magnetic resonance in one and two dimensions, Clarendon press, ISBN 9780198556473, 1992. a
Fugariu, I., Bermel, W., Lane, D., Soong, R., and Simpson, A. J.: InPhase Ultra HighResolution In Vivo NMR, Angew. Chem.Int. Edit., 56, 6324–6328, https://doi.org/10.1002/anie.201701097, 2017. a
He, Q., Richter, W., Vathyam, S., and Warren, W.: Intermolecular MultipleQuantum Coherences and Cross Correlations in Solution NuclearMagneticResonance, J. Chem. Phys., 98, 6779–6800, https://doi.org/10.1063/1.464770, 1993. a
Huang, Y., Cai, S., Chen, X., and Chen, Z.: Intermolecular singlequantum coherence sequences for highresolution NMR spectra in inhomogeneous fields, J. Magn. Reson., 203, 100–107, https://doi.org/10.1016/j.jmr.2009.12.007, 2010. a, b
Klages, J., Kessler, H., Glaser, S. J., and Luy, B.: JONLYTOCSY: Efficient suppression of RDCinduced transfer in homonuclear TOCSY experiments using JESTER1derived multiple pulse sequences, J. Magn. Reson., 189, 217–227, https://doi.org/10.1016/j.jmr.2007.09.010, 2007. a
Kramer, F. and Glaser, S. J.: Efficiency of homonuclear HartmannHahn and COSYtype mixing sequences in the presence of scalar and residual dipolar couplings, J. Magn. Reson., 155, 83–91, https://doi.org/10.1006/jmre.2002.2511, 2002. a
Kramer, F., Peti, W., Griesinger, C., and Glaser, S. J.: Optimized homonuclear CarrPurcelltype dipolar mixing sequences, J. Magn. Reson., 149, 58–66, https://doi.org/10.1006/jmre.2000.2271, 2001. a
Kupce, E. and Freeman, R.: Adiabatic pulses for wideband inversion and broadband decoupling, J. Magn. Reson. A, 115, 273–276, https://doi.org/10.1006/jmra.1995.1179, 1995. a
Lee, S., Richter, W., Vathyam, S., and Warren, W. S.: Quantum treatment of the effects of dipoledipole interactions in liquid nuclear magnetic resonance, J. Chem. Phys., 105, 874–900, https://doi.org/10.1063/1.471968, 1996. a, b, c, d
Lin, Y., Huang, Y., Cai, S., and Chen, Z.: Intermolecular Zero Quantum Coherence in NMR Spectroscopy, in: Annual Reports on Nmr Spectroscopy, Elsevier Academic Press Inc, San Diego, edited by: Webb, G. A., 78, 209–257, https://doi.org/10.1016/B9780124047167.000055, 2013. a
Nagayama, K., Wuthrich, K., and Ernst, R.: 2Dimensional spinecho correlated spectroscopy (SECSY) for H1NMR studies of biological macromolecules, Biochem. Biophys. Res. Commun., 90, 305–311, https://doi.org/10.1016/0006291X(79)916255, 1979. a
Pelupessy, P.: Radiation damping strongly perturbs remote resonances in the presence of homonuclear mixing, Magn. Res., 3, 43–51, https://doi.org/10.5194/mr3432022, 2022a. a
Pelupessy, P.: Transfer of phase coherence by the dipolar field in total correlation liquid state nuclear magnetic resonance spectroscopy, J. Chem. Phys., 157, 164202, https://doi.org/10.1063/5.0120909, 2022b. a, b, c, d, e, f, g, h, i, j, k, l, m, n
Piotto, M., Saudek, V., and Sklenar, V.: GradientTailored Excitation for SingleQuantum Nmr–Spectroscopy of Aqueous–Solutions, J. Biomol. NMR, 2, 661–665, https://doi.org/10.1007/BF02192855, 1992. a
Rucker, S. and Shaka, A.: BroadBand Homonuclear Cross Polarization in 2d Nmr Using Dipsi2, Mol. Phys., 68, 509–517, https://doi.org/10.1080/00268978900102331, 1989. a
Shaka, A., Keeler, J., and Freeman, R.: Evaluation of a New BroadBand Decoupling Sequence – Waltz16, J. Magn. Reson., 53, 313–340, https://doi.org/10.1016/00222364(83)900355, 1983. a
Shaka, A., Barker, P., and Freeman, R.: ComputerOptimized Decoupling Scheme for Wideband Applications and LowLevel Operation, J. Magn. Reson., 64, 547–552, https://doi.org/10.1016/00222364(85)901222, 1985. a
Suryan, G.: Nuclear Magnetic Resonance and the Effect of the Methods of Observation, Curr. Sci., 18, 203–204, 1949. a
Vathyam, S., Lee, S., and Warren, W. S.: Homogeneous NMR spectra in inhomogeneous fields, Science, 272, 92–96, https://doi.org/10.1126/science.272.5258.92, 1996. a, b
Warren, W., Richter, W., Andreotti, A., and Farmer, B.: Generation of Impossible CrossPeaks Between Bulk Water and Biomolecules in Solution Nmr, Science, 262, 2005–2009, https://doi.org/10.1126/science.8266096, 1993. a, b
Warren, W. S., Lee, S., Richter, W., and Vathyam, S.: Correcting the classical dipolar demagnetizing field in solution NMR, Chem. Phys. Lett., 247, 207–214, https://doi.org/10.1016/00092614(95)011845, 1995. a
Zangger, K. and Sterk, H.: Homonuclear broadbanddecoupled NMR spectra, J. Magn. Reson., 124, 486–489, https://doi.org/10.1006/jmre.1996.1063, 1997. a